A fundamental quest in the theory of computing is to understand the power of randomness. It is not known whether every problem with an efficient randomized algorithm also has one that does not use randomness. One of the extensively studied problems under this theme is that of perfect matching. The perfect matching problem has a randomized parallel (NC) algorithm based on the Isolation Lemma of Mulmuley, Vazirani, and Vazirani. It is a long-standing open question whether this algorithm can be derandomized. In this article, we give an almost complete derandomization of the Isolation Lemma for perfect matchings in bipartite graphs. This gives us a deterministic parallel (quasi-NC) algorithm for the bipartite perfect matching problem.
Derandomization of the Isolation Lemma means that we deterministically construct a weight assignment so that the minimum weight perfect matching is unique. We present three different ways of doing this construction with a common main idea.
A perfect matching in a graph is a subset of edges such that every vertex has exactly one edge incident on it from the subset (Figure 1). The perfect matching problem, PM, asks whether a given graph contains a perfect matching. The problem has played an important role in the study of algorithms and complexity. The first polynomial-time algorithm for the problem was given by Edmonds,7 which, in fact, motivated him to propose polynomial time as a measure of efficient computation.
Figure 1. A graph, with the bold edges showing a perfect matching.
Perfect matching was also one of the first problems to be studied from the perspective of parallel algorithms. A parallel algorithm is one where we allow use of polynomially many processors running in parallel. And to consider a parallel algorithm as efficient, we require the running time to be much smaller than a polynomial. In particular, the complexity class NC is defined as the set of problems which can be solved by a parallel computer with polynomially many processors in poly-logarithmic time.
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